178 BOOK OF MATHEMATICAL PROBLEMS. 



829. being a fixed point on a conic, OP, OQ any two 

 chords, OR the chord normal at 0; prove that there exists a 

 straight line passing through the pole of OR, such that the 

 tangents at F, Q intercept on it a length which subtends at an 

 angle twice POQ. 



830. On any straight line can be found two points, conjugate 

 to a given conic, such that the distance between them subtends a 

 right angle at a given point. 



831. ABC is a triangle, any point, and straight lines 

 through at right angles to OA, OB, 00 meet the respectively 

 opposite sides in A', 2?, C? ; prove that any conic which touches 

 the sides of the triangle and the straight line A'B'C' will subtend 

 a right angle at 0, 



832. An ellipse is described about an acute-angled triangle 

 A BC, and one focus is the centre of perpendiculars of the triangle ; 

 prove that its latus rectum is 



cos A cos B cos C 



2R 



A . . C' 

 2" sm 7 sm 2 



833. A parabola and hyperbola have a common focus and 

 axis, and the parabola touches the directrix of the hyperbola ; 

 prove that any straight line through the focus is harmonically 

 divided by any tangent to the parabola and the two parallel 

 tangents to the hyperbola. 



834. A series of conies are described having equal latera 

 recta, the focus of a given parabola their common focus, and 

 tangents to the parabola their directrices; prove that the com- 

 mon tangents of any two intersect on the directrix of the 

 parabola. 



835. With a given point as focus four conies can be drawn 

 circumsrribing a given triangle, and the latus rectum of one of 

 them will be equal to the sum o' % those of the other three. Also, 

 if any conic A be drawn touching the directrices of the four 



